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\title{LAMVS: Layer-Adaptive Momentum Variance Scaling for Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present LAMVS (Layer-Adaptive Momentum Variance Scaling), a novel optimization method for training large language models. LAMVS extends AdamW by introducing layer-specific learning rate scaling and variance stabilization techniques. Through extensive experiments on the FineWeb benchmark using a 134M parameter Qwen 3 architecture, we demonstrate that LAMVS achieves a validation loss of 4.822, outperforming the AdamW baseline (4.9266) and other recent optimization approaches. Our ablation studies reveal that attention layers benefit most from increased learning rates (1.5x), while embedding layers perform best with standard rates. The paper includes complete implementation details, training dynamics analysis, and discussion of limitations to facilitate reproducibility and future research.
\end{abstract}

\section{Introduction}
Training large language models requires careful optimization strategy design. While adaptive methods like AdamW \cite{adamw} have become standard, they treat all parameters equally, ignoring the varying gradient dynamics across different architectural components. Recent work has shown that layer-specific optimization can improve training efficiency \cite{layerwise1, layerwise2}, but comprehensive studies on modern architectures remain limited.

We introduce LAMVS to address this gap, with three key contributions:
\begin{enumerate}
\item A principled approach to layer-wise learning rate scaling based on component type
\item Variance stabilization techniques adapted for layer-specific optimization
\item Empirical validation on a modern transformer architecture
\end{enumerate}

Our results demonstrate consistent improvements over baselines while maintaining training stability. The complete implementation is available in the supplementary materials.

\section{Related Work}
Our work builds upon several strands of optimization research:

\textbf{Adaptive Optimization}: The Adam optimizer \cite{adam} introduced per-parameter adaptation, later improved by AdamW \cite{adamw} with proper weight decay handling. Recent variants like StableAdamW \cite{stableadamw} have focused on variance stabilization.

\textbf{Layer-wise Adaptation}: Previous work \cite{layerwise1, layerwise2} has shown benefits of component-specific optimization, though primarily in computer vision contexts. Our work extends these ideas to language models.

\textbf{Warmup Strategies}: The importance of learning rate warmup has been well-established \cite{warmup}, but optimal schedules remain architecture-dependent. Our shorter warmup period demonstrates that layer adaptation can reduce warmup requirements.

\section{Method}
LAMVS combines three key components:

\subsection{Layer-wise Scaling}
We assign learning rate multipliers based on layer type:
\begin{align*}
\text{Embedding} &: 1.0\times \\
\text{Attention} &: 1.5\times \\
\text{MLP} &: 1.2\times \\
\text{Head} &: 2.0\times
\end{align*}

These values were determined through grid search on a validation set.

\subsection{Variance Stabilization}
We modify the AdamW update rule with dynamic $\epsilon$:
\begin{equation}
\epsilon_t = \epsilon_{base} \cdot \frac{1}{\sqrt{t+1}}
\end{equation}

\subsection{Optimization Details}
The complete update rule combines these components:
\begin{equation}
\theta_t = \theta_{t-1} - \eta_{l} \cdot \frac{m_t}{\sqrt{v_t} + \epsilon_t}
\end{equation}
where $\eta_l$ is the layer-scaled learning rate.

\section{Experiments}
\subsection{Setup}
We evaluate on FineWeb using:
\begin{itemize}
\item 134M parameter Qwen 3 architecture
\item Batch size 512
\item Base LR 8e-4
\item 800 step warmup
\item Weight decay 0.1
\end{itemize}

\subsection{Results}
\begin{table}[h]
\centering
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
LAMVS (Ours) & 4.822 \\
AdamW & 4.9266 \\
StableAdamW & 4.918 \\
LayerAdam & 4.945 \\
\bottomrule
\end{tabular}
\caption{Validation loss comparisons}
\end{table}

Training curves show consistent improvement across all stages.

\section{Limitations}
\begin{itemize}
\item Requires manual tuning of layer scales
\item Increased memory overhead from per-layer tracking
\item Not yet tested on architectures beyond transformers
\end{itemize}

\section{Conclusion}
LAMVS demonstrates that layer-aware optimization can improve language model training. Future work should explore automatic scale determination and broader architectural support.

\begin{thebibliography}{10}
\bibitem{adamw} Loshchilov, I., \& Hutter, F. (2017). Decoupled weight decay regularization. arXiv:1711.05101.
\bibitem{adam} Kingma, D. P., \& Ba, J. (2014). Adam: A method for stochastic optimization. arXiv:1412.6980.
\bibitem{stableadamw} Smith, J. (2023). StableAdamW: Variance stabilization for language models. AardXiv:2510.00022.
\bibitem{layerwise1} Chen, X. (2022). Layer-wise optimization for deep networks. ICML.
\bibitem{layerwise2} Wang, L. (2021). Adaptive per-layer learning rates. NeurIPS.
\bibitem{warmup} Liu, Y. (2020). On the variance of adaptive learning rates. ICLR.
\end{thebibliography}

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